The purple graph is a transformation of \(y=f(x)\) and this transformation involves three parameters, \(a\), \(b\) and \(c\). The parameters are related to scaling and translation, but the numerical value of the parameter \(b\) is not necessarily the same as the value the graph is translated by along the \(x\)-axis.

You may like to explore what the translation is for the same value of \(b\) but different values of \(a.\)

What if we applied these ideas to the orange curve too?

Now try to evaluate each of the following integrals.

\(\int_{-4}^{-2}\sqrt{x+5}\,dx\)

\(\int_{0.2}^{1.3}e^{2x}\,dx\)

Think about what is happening in these GeoGebra files as you move the sliders.